Chem. Mater. 2006, 18, 3579-3584
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Bond-Curvature Effect of Sidewall [2+1] Cycloadditions of Single-Walled Carbon Nanotubes: A New Criterion To the Adduct Structures Junqian Li,*,†,‡ Guixiao Jia,† Yongfan Zhang,† and Yong Chen† Department of Chemistry, Fuzhou UniVersity, Fuzhou, Fujian, 350002, China, and State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, The Chinese Academy of Sciences, Fuzhou, Fujian, 350002, China ReceiVed March 9, 2006. ReVised Manuscript ReceiVed May 19, 2006
A unique property, bond curvature K, is proposed as a universal criterion for structures and reactivities of sidewall [2+1] cycloadditions of SWCNTs. B3LYP/6-31G* calculations for cycloadditions on various types of SWCNTs show that the binding energies of the open structures and the changes in C-C bond lengths of the 3MR structures increase linearly with K. When K is large, the open structure is formed, whereas when K is small, the formation of the configuration with a three-membered ring (3MR) is favorable. A boundary of K for producing different structures is about 1.5 nm-1 for the tubes with moderate radii. The cycloaddition of CCl2 on any C-C bond of zigzag tubes with smaller K values will lead to adducts with 3MR structures, which clarifies the contradiction between the experimental phenomenon and previous theoretical predictions.
1. Introduction Single-walled carbon nanotubes (SWCNTs) exhibit interesting electronic, mechanical, and structural properties,1,2 making them promising for applications in various fields such as chemical sensors and nanometer-scale electronic devices.3-6 The experimental and theoretical investigations show that the sidewall of SWCNT has a higher curvature-induced chemical reactivity than that of the flat graphite.7-9 Consequently, the chemical functionalization of open ends and sidewalls of SWCNTs would play a vital role in tailoring their properties. One of the most-promising approaches to purifying SWCNTs and expanding their application areas is the chemical derivatizations.10-14 Among these, covalent * Corresponding author. E-mail:
[email protected]. † Fuzhou University. ‡ Fujian Institute of Research on the Structure of Matter, The Chinese Academy of Sciences.
(1) Dresselhaus, M. S.; Dresselhaus, G.; Avouris, P. Carbon Nanotubes; Springer: Berlin, 2001. (2) Dai, H. Surf. Sci. 2002, 500, 218-241. (3) Collins, P. G.; Bradley, K.; Ishigami, M.; Zettl, A.; Science 2000, 287, 1801-1804. (4) Kong, J.; Franklin, N. R.; Zhou, C.; Chapline, M. G.; Peng, S.; Cho, K.; Dai, H. Science 2000, 287, 622-625. (5) Baughman, R. H.; Zakhidov, A. A.; de Heer, W. A. Science 2002, 297, 787-792. (6) Ajayan, P. M. Chem. ReV. 1999, 99, 1787-1800. (7) Niyogi, S.; Hamon, M. A.; Hu, H.; Zhao, B.; Bhowmik, P.; Sen, R.; Itkis, M. E.; Haddon, R. C. Acc. Chem. Res. 2002, 35, 1105-1113. (8) Tada, K.; Furuya, S.; Watanabe, K. Phys. ReV. B 2001, 63, 155405155408. (9) Gu¨lseren, O.; Yildirim, T.; Ciraci, S. Phys. ReV. Lett. 2001, 87, 116802-116805. (10) Chen, J.; Hamon, M. A.; Hu, H.; Chen, Y.; Rao, A. M.; Eklund, P. C.; Haddon, R. C. Science 1998, 282, 95-98. (11) Wong, S. S.; Joselevich, E.; Woolley, A. T.; Cheung, C. L.; Lieber, C. M. Nature 1998, 394, 52-55. (12) Mickelson, E. T.; Chiang, I. W.; Zimmerman, J. L.; Boul, P. J.; Lozano, J.; Liu, J.; Smalley, R. E.; Hauge, R. H.; Margrave, J. L. J. Phys. Chem. B 1999, 103, 4318-4322.
sidewall functionalizations, especially the [2+1] cycloaddition of C-C bond on the tube wall, may increase their solubility15-19 and improve their bioactivity and biocompatibility etc.20,21 Therefore, the investigation for the [2+1] cycloaddition has been an important subject for the covalent sidewall chemistry of carbon nanotubes.22-24 The [2+1] cycloadditions of dichlorocarbene CCl2,10,25,26 Bingel (COOEt)2C, and its derivations,27 as well as (R)oxycarbonylaziridino28 have been achieved experimentally. According to the dramatic changes in the optical spectra of SWCNTs, Hu26 et al. considered the local structures of adducts attached to dichlorocarbene to be three-membered (13) Riggs, J. E.; Guo, Z.; Carroll, D. L.; Sun, Y.-P. J. Am. Chem. Soc. 2000, 122, 5879-5880. (14) Chen, Q.; Dai, L.; Gao, M.; Huang, S.; Mau, A. J. Phys. Chem. B 2001, 105, 618-622. (15) Sun, Y.-P.; Fu, K.; Lin, Y.; Huang, W. Acc. Chem. Res. 2002, 35, 1096-1104. (16) Dyke, C. A.; Tour, J. M. J. Phys. Chem. A 2004, 108, 11151-11159. (17) Banerjee, S.; Hemraj-Benny, T.; Wong, S. S. AdV. Mater. 2005, 17, 17-29. (18) Hirsch, A.; Vostrowsky, O. Top. Curr. Chem. 2005, 245, 193-238. (19) Georgakilas, D.; Kordatos, K.; Prato, M.; Guldi, D. M.; Holzinger, M.; Hirsch, A. J. Am. Chem. Soc. 2002, 124, 760-761. (20) Pantarotto, D.; Partidos, C. D.; Graff, R.; Hoebeke, J.; Briand, J.-P.; Prato, M.; Bianco, A. J. Am. Chem. Soc. 2003, 125, 6160-6164. (21) Alvaro, M.; Atienzar, P.; de la Cruz, P.; Delgado, J. L.; Garcia, H.; Langa, F. J. Phys. Chem. B 2004, 108, 12691-12697. (22) Chen, Z. F.; Nagase, S.; Hirsch, A.; Haddon, R. C.; Thiel, W.; Schleyer, P. R. Angew. Chem., Int. Ed. 2004, 43, 1552-1554. (23) Barone, V.; Heyd, J.; Scuseria, G. E. Chem. Phys. Lett. 2004, 389, 289-292. (24) Lu, X.; Chen, Z. F. Chem. ReV. 2005, 105, 3643-3696. (25) Kamaras, K.; Itkis, M. E.; Hu, H.; Zhao, B.; Haddon, R. C. Science, 2003, 301, 1501-1501. (26) Hu, H.; Zhao, B.; Hamon, M. A.; Kamaras, K.; Itkis, M. E.; Haddon, R. C. J. Am. Chem. Soc. 2003, 125, 14893-14900. (27) Coleman, K. S.; Bailey, S. R.; Fogden, S.; Green, M. L. H. J. Am. Chem. Soc. 2003, 125, 8722-8723. (28) Holzinger, M.; Abraham, J.; Whelan, P.; Graupner, R.; Ley, L.; Hennrich, F.; Kappes, M.; Hirsch, A. J. Am. Chem. Soc. 2003, 125, 8566-8580.
10.1021/cm060563v CCC: $33.50 © 2006 American Chemical Society Published on Web 06/23/2006
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rings (3MR) with a cyclopropane configuration. So far, there have been several reports regarding theoretical investigations of the additions of oxygen,22-24,29-32 carbene,22,32 dichlorocarbene,33,34 and nitrene22,35 on the sidewall of SWCNTs. However, in terms of the prediction for ultimate structures, some controversies still exist. Lu36 et al. showed by using ONIOM method that when dichlorocarbene, silylene, germylene, and oxycarbonylnitrene were added to the sidewall of an armchair (5,5) SWCNT, epoxide-like structures of adducts were formed. Recently, by the full optimization for the structure of SWCNTs on the basis of the critical theoretical method, Chen et al. suggested that the breaking of conjointed C-C bond is possible and that the epoxidelike structures are not energetically favorable.22 The results obtained from the theoretical calculations on the B3LYP/6-31G* level have a higher reliability;22 however, several cruxes related to the [2+1] cycloadditions are still puzzling. For example, why are the structures of [2+1] adducts derived from experiments different from those predicted theoretically?22,34 How do we find a proper index that predicts the reactivities and adduct structures, etc.? Many authors take the strength of the C-C bond,22 the tube diameter (or tubular curvature),32 the pyramidalization angle (θp), and the π-orbital misalignment angle (φ)7-9 as criteria for predictingt the configurations of [2+1] adducts; however, these criteria are not universal. In the present work, we will see that, in many cases, the short (or more double-bond character22) C-C bonds favor the formation of 3MR structures; on the contrary, the long C-C bonds are broken, which is where the opened structure is formed during the [2+1] cycloaddition. The tubular diameter is also not a good criterion, because it cannot distinguish the different C-C bonds in a given SWCNT and can be used only for the comparisons of the adduct structures involving the C-C bonds with θ ) 90° (θ is the C-C bond obliquity angle, as presented in Figure 1) in different SWCNTs. Although the values of θp and φ have been proven to be a useful index for describing the local reactivity of the fullerenes,7-9 they are not perfect parameters for the SWCNTs, because it is impossible to estimate an opened cycloaddition structure with the broken C-C σ bond. Therefore, finding a suitable criterion to discriminate the chemical reactivities and structures of the same or different SWCNTs is still a problem awaiting a solution. In practice, the precise characterization of the structure of the adduct has become a bottleneck for the study of sidewall cycloaddition.22,24 In addition, up to now, there is still a lack of systemic and critical theoretical (29) Sorescu, D. C.; Jordan, K. D.; Avouris, P. J. Phys. Chem. B 2001, 105, 11227-11232. (30) Lu, X.; Zhang, L.; Xu, X.; Wang, N.; Zhang, Q. J. Phys. Chem. B 2002, 106, 2136-2139. (31) Dag, S.; Gu¨lseren, O.; Yildirim, T.; Ciraci, S. Phys. ReV. B 2003, 67, 165424/1-10. (32) Lu, J.; Nagase, S.; Zhang, X.; Maeda, Y.; Wakahara, T.; Nakahodo, T.; Tsuchiya, T.; Akasaka, T.; Yu, D.; Gao, Z.; Han, R.; Ye, H. J. Mol. Struct. (THEOCHEM) 2005, 725, 255-257. (33) Li, R. F.; Shang, Z. F.; Wang, G. C.; Pan, Y. M.; Cai, Z. S.; Zhao, X. Z. J. Mol. Struct. (THEOCHEM) 2002, 583, 241-247. (34) Zhao, J. J.; Chen, Z. F.; Zhou, Z.; Park, H.; Schleyer, P. R.; Lu, J. P. Chem. Phys. Chem. 2005, 6, 598-601. (35) Holzinger, M.; Vostrowsky, O.; Hirsch, A.; Hennrich, F.; Kappes, M.; Weiss, R.; Jellen, F. Angew. Chem., Int. Ed. 2001, 40, 4002-4005. (36) Lu, X.; Tian, F.; Zhang, Q. J. Phys. Chem. B 2003, 107, 8388-8391.
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Figure 1. Nonequivalent C-C bonds in the (a) chiral, (b) zigzag, and (c) armchair SWCNTs ; X expresses species in [2+1] cycloadditions.
investigations for [2+1] cycloadditions of SWCNTs with different types and sizes, especially for the chiral SWCNTs that possess the most common SWCNT structure.37 Here, we report a critical theoretical study on the basis of the first principle method for the structures, energies, and their varying rules of [2+1] sidewall cycloadditions on all kinds of the SWCNTs, including the chiral, armchair, and zigzag tubes. A new criterion, bond curvature, which can estimate the [2+1] adduct structures for both the same and different SWCNTs, is proposed in this work, and the contradiction of the cycloaddition structures predicted by the experiments and the theoretical calculations is elucidated. 2. Definition of Bond Curvature and Computational Model Figure 1 shows the nonequivalent C-C bonds of different types of SWCNTs. θ is an acute angle between the tubular axis and the C-C bond on the plane of the unfolded SWCNT, defined as the C-C bond obliquity angles or the bond angles. The minimum θmin is just the chiral angle of a SWCNT, and the other C-C bond angles are equal to 60° - θmin and 60° + θmin. For the [2+1] cycloaddition, the curvature of SWCNT32 (or point curvature) that expresses only the curvature of every carbon atom on the tubular circumference does not exactly reflect the strain energies of various C-C bonds. Bond curvature, a new indicator proposed in this work, may effectively estimate the strain energy of a certain C-C bond, and the [2+1] cycloaddition is just sensitive to the C-C bond curvature. Bond curvature K is defined as the average curvature of the corresponding arc of a C-C bond in the SWCNT. As shown in Figure 2, the terminal tangents of the arcs of the C1-C2 bond and its projected bond C′1-C2 on the tubular circumference lie in planes M and N, respectively. The formula of bond curvature K of the C1-C2 bond can be derived from the curvature of arc C′ k 1C2 on the circumference with tubular radius R (see part 1 of the Supporting Information for details). K)
φ C k 1C2
)
φ' sin2θ C′ k 1 C2
)
sin2θ R
From the above formula of K, it can be seen that K is related to the values of θ and R. For the C-C bonds of a given SWCNT, K changes only with θ, and for the bonds with the same θ among different SWCNTs, K changes with R. K relates only to a given C-C bond in this work, so it can express the strain energy Es of (37) Miyauchi, Y.; Chiashi, S.; Murakami, Y.; Hayashida, Y.; Maruyama, S. Chem. Phys. Lett. 2004, 387, 198-203.
Bond-CurVature Effect of [2+1] Cycloadditons of SWCNTs
Chem. Mater., Vol. 18, No. 15, 2006 3581 the binding energies Eb(I) can be rewritten as the following simplified expression Eb(I) ) EX + EC-C + ES + E′SWCNT - (2EC-X(I) + EX + E′SWCNT) ) EC-C + ES - 2EC-X (I) where EC-C and EC-X are the binding energies of the C-C bond and the C-X bond, respectively; ES is the strain energy of the C-C bond; and E′SWCNT is the energy of a fictitious SWCNT from which the C-C bond participating in the [2+1] cycloaddition is removed. Because the values of EC-C, EC-X, and their different ∆E ) EC-C - 2EC-X are fixed, it could be expected that there is a linear relationship between Eb(I) and ES or K, namely Eb(I) ) ∆E + aK. For the binding energy of the formation of type II adduct Eb(II) ) (EC-C - E′C-C) + (ES - E′S) - 2EC-X
Figure 2. Three-dimensional sketch map for the definition of the bond curvature (K); also see part 1 of the Supporting Information for details.
the C-C bond,38 namely ES ) aK, where a is a proportionality factor. The binding energies Eb for the [2+1] cycloaddition can be calculated according to the following expression Eb ) Etube + EX - Etube-X
(II)
where E′C-C and E′S are the C-C bond energy and the strain energy of the adduct, respectively. |2EC-X(I)| > |2EC-X(II)|and (EC-C E′C-C) + (ES - E′S) ≈ 0 because of their counteraction, so in general, Eb(II) < Eb(I), and there is not a simple relationship between Eb(II) and K. A linear relationship may exist between the change in the C-C bond length (∆RC-C(II)) and K, because the value of K will increase along with a reduction in the C-C bond length.
3. Results and Discussion where Etube-x is the total energy of the tube-adsorbate system and EX (X ) O, CH2, NH, SiH2, or CCl2, etc.) and Etube are the energies of an isolated X species and the pristine tube, respectively. The optimized structures and binding energies were obtained by the hybrid density functional theory (DFT), namely Becke’s three parameter hybrid functional,39 B3LYP method was employed in the present work. For all atoms considered, the standard 6-31G* basis sets were used. All the calculations were performed with the Gaussian 03 package.40 On the basis of the method mentioned above, we obtain a series of results of various O/SWCNT derivatives, and CH2, NH, and SiH2 adducts for (6,2) and (6,3) SWCNTs, as well as CCl2 adducts for (20,0) and (22,0) SWCNTs. During the calculations, the SWCNTs are simulated by a section of the tube, and the dangling C-C bonds at the two ends of the fragment are saturated by H atoms. For the chiral SWCNTs, one unit cell is selected except for the (6,5) tube, for which a fragment with 100 carbon atoms is chosen because there are too many carbon atoms in the unit cell. In [2+1] cycloadditions, the C-C bond a, b, or c in the middle-exohedral sidewalls of the SWCNTs (Figure 1) is considered. For example, the C-C bonds a, b, and c of the chiral SWCNT (6,3)(C78H18) (Figure 1a), the C-C bonds a and b of the zigzag (8,0)(C96H16) (Figure 1b) and the armchair (5,5)(C110H20) (Figure 1c) participated in the cycloadditions (also see Tables 1 and 2). Furthermore, to investigate the adsorptions of O atom on the periodical zigzag SWCNTs, we also have carried out some additional calculations within the framework of DFT with a plane wave basis set and pseudopotentials (see the Supporting Information for the details of the calculations). The cycloaddition of O, CH2, NH, SiH2, and CCl2 on the sidewall of SWCNT may result in two types of the C-C bonds on the reaction site, namely the broken C-C bond in the opened C-X-C configuration and the C-C bond in the closed 3MR formed with X, which are denoted by type I and type II, respectively, in this work. For the cycloaddition of the formation of the type I adduct, (38) Robertson, D. H.; Brenner, D. W.; Mintmire, J. W. Phys. ReV. B 1992, 45, 12592-12595. (39) Becke, A. D. J. Chem. Phys. 1993, 98, 5648-5652. (40) Frisch, M. J.; et al.; Gaussian 03, Gaussian Inc.: Pittsburgh, PA, 2003.
3.1. X/SWCNT System with Moderate R. Because SWCNTs used in the chemical modifications are mainly HiPCO tubes with moderate diameter (7-14 Å), the cycloaddition rule of these SWCNTs will be especially considered. On the basis of the calculated results for various C-C bonds of all types of SWCNTs, we established two linear relationships between Eb(I) and K and ∆RC-C(II) and K (Figures 3 and 4, or Tables 1 and 2). However, ∆RC-C(I) of the type I bond and Eb(II) of the formation of the type II adduct vary irregularly with increasing K (Tables 1 and 2). From Tables 1 and 2, it can be seen that Eb(I) is larger than Eb(II) as a whole and that the larger K, the easier the C-C bond is broken. A boundary value of K for determining whether the C-C bond is broken or not is obtained, namely K ≈ 1.5 nm-1. Hence, it is favorable to form the type I adduct as K > 1.5 nm-1, whereas the type II structure is obtained as K < 1.5 nm-1. It is worth noting that, according to our results, whether the C-C bond is broken or not is not directly relative to the bond length or strength. As an example, for the cycloaddition of O on a (6,1) tube, the C-C bond with a short length of 1.414 Å (K ) 0.068 nm-1) is not broken, and a C-C bond with a long distance of 1.439 Å (K ) 2.429 nm-1) is destroyed. On the other hand, there also exist cases in which the short C-C bond is broken whereas the long C-C bond is kept. For instance, in the (6,3) tube, the C-C bond with a length of 1.435 Å (K ) 0.344 nm-1) is preserved; however, another strong C-C bond with a length of 1.429 Å (K ) 3.101 nm-1) is broken. It is also obvious that the tube diameter is not a good index for judging whether the C-C bond is broken. A typical example is that in a given SWCNT, such as the (6,1) tube mentioned above, the breaking and nonbreaking of C-C bonds can coexist. Another example is that, although the tube diameter of the (6,2) tube (0.564 nm) is smaller than that of
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Table 1. Tubular Radius (R), Bond Angles (θ), Bond Curvature (K), Lengths of C-C Bonds before (R0c-c) and after (R′c-c) Cycloaddition, Changes in C-C Bond Lengths (∆Rc-c), and Binding Energies (Eb) of the Formation of the Type I Adduct for the Cycloaddition of the O Atom on Various SWCNTs
(12,0) (C120H24) (6,2) (C100H16) (10,0) (C100H20) (8,0) (C96H16) (6,1) (C168H14) (6,6) (C132H24) (6,5) (C100H22) (6,4) (C148H20) (5,5) (C110H20) (6,3) (C78H18) (6,0) (C84H12) (6,2) (C100H16) (6,1) (C168H14) (4,4) (C88H16)
R (nm)
θ (deg)
K (nm-1)
R0c-c (Å)
R′c-c (Å)
∆Rc-c(I) (Å)
Eb(I) (kcal/mol)
0.470 0.282 0.392 0.313 0.257 0.407 0.373 0.341 0.339 0.311 0.235 0.282 0.257 0.271
60.00 46.10 60.00 60.00 52.41 90.00 87.00 83.41 90.00 79.11 60.00 73.90 67.59 90.00
1.596 1.841 1.913 2.396 2.429 2.457 2.673 2.894 2.950 3.101 3.191 3.273 3.325 3.690
1.422 1.433 1.425 1.428 1.439 1.432 1.430 1.421 1.437 1.429 1.443 1.431 1.442 1.447
2.097 2.123 2.109 2.125 2.154 2.118 2.111 2.096 2.127 2.114 2.179 2.141 2.177 2.139
0.675 0.690 0.684 0.697 0.715 0.686 0.681 0.675 0.690 0.685 0.736 0.710 0.735 0.692
57.03 58.93 62.63 69.87 70.28 73.95 76.03 76.36 82.39 81.17 81.35 82.98 85.25 94.91
Table 2. Tubular Radius (R), Bond Angles (θ), Bond Curvature (K), Lengths of C-C Bonds before (R0c-c) and after (R′c-c) Cycloaddition, Changes in Adduct Lengths (∆Rc-c), and Binding Energies (Eb) of the Formation of the Type II C-C Bond for the Cycloaddition of the O Atom on Various SWCNTs
(6,0) (C84H12) (6,1) (C168H14) (6,2) (C100H16) (6,3) (C78H18) (6,4) (C148H20) (6,5) (C100H22) (6,6) (C120H24) (6,5) (C100H22) (6,4) (C148H20) (6,3) (C78H18)
R (nm)
θ (deg)
K (nm-1)
R0c-c (Å)
R′c-c (Å)
∆Rc-c(II) (Å)
Eb(II) (kcal/mol)
0.235 0.257 0.282 0.311 0.341 0.373 0.407 0.373 0.341 0.311
0 7.59 13.90 19.11 23.41 27.00 30.00 33.00 36.59 40.89
0 0.068 0.205 0.344 0.463 0.553 0.614 0.795 1.042 1.378
1.416 1.414 1.428 1.435 1.430 1.423 1.417 1.424 1.435 1.421
1.469 1.470 1.490 1.500 1.506 1.506 1.507 1.518 1.541 1.528
0.053 0.056 0.062 0.065 0.076 0.083 0.090 0.094 0.106 0.107
72.58 64.56 49.57 42.02 46.13 47.97 51.26 48.42 45.18 56.15
Figure 3. Variation in the binding energy (Eb) as a function of bond curvature for the formation of the type I adduct during the cycloaddition of the O atom.
Figure 4. Variation in the change of C-C bond length (∆RC-C) as a function of bond curvature for the formation of the type II adduct during the cycloaddition of the O atom.
the (6,4) tube (0.682 nm), the C-C bond of the former (K ) 0.205 nm-1) is kept but the C-C bond of the latter (K ) 2.894 nm-1) is broken. In the research,7-9 authors suggest that the magnitude of the misalignment angle for π-orbital overlap, φ, could reflect the SWCNT’s reactivity. However, there exists a contradiction between the φ value of a C-C bond of an armchair tube along the tubular circumference and the reactivity, because the φ value is zero but the reactivity of this C-C bond is strongest in the armchair tube, as seen in this work and others.22,32 Furthermore, the cycloaddition of the O atom at the C-C bond with θ ) 90° in the armchair SWCNT (5,5) shows an opened structure, whereas the 3MR is produced when the cycloaddition takes place at the other
C-C bonds with θ ) 30°, which implies that the C-C bond with θ ) 90° exhibits a higher [2+1] cycloaddition reactivity. This conclusion contradicts the result deduced from the criterion of φ, because the φ value of the C-C bond with θ ) 90° is equal to zero and is obviously less than that of the other C-C bond with θ ) 30°.7-9 Moreover, in general, the C-C bond with θ ) 60° in a zigzag SWCNT (10,0) forms an open structure and the C-C bond with θ ) 30° in the (5,5) SWCNT tends to create 3MR structures; however, the values of θp and φ7-9 of the former are less than those of the latter. Therefore, the parameter φ, which is localized and does not consider σ-bonding, is obviously difficult to determine for the adduct structures and reactivities on SWCNTs.
Bond-CurVature Effect of [2+1] Cycloadditons of SWCNTs
Figure 5. Variations in the binding energy (Eb) for the formation of the type I adduct (solid line) and the change in C-C bond length (∆RC-C) for the formation of the type II adduct (dashed line) as functions of bond curvature for the cycloadditions of CH2, SiH2, and NH on (6,2) (denoted by filled circles) and (6,3) tubes (denoted by hollow circles).
According to the structural character of SWCNTs, K involving the two parameters R and θ can solely determine the types of the C-C bonds and predict the reactivities of C-C bonds and adduct structures for both the same and different SWCNTs, so it is universal. In practice, both π-delocalized and σ-bonding interactions of the C-C bond are considered in K, so K along the tubular circumference of the armchair tube is maximal. Of course, it exhibits a high chemical reactivity. Therefore, compared with the above criteria that have been usually adopted in many studies, K is more suitable for estimating the reactivity and structure of the SWCNT. Herein, what is meant by the high reactivity is the favorable reaction in binding energy, namely thermodynamically, as in many previous works.7-9,24 Of course, there may be various kinetic mechanisms of the reaction for different types of SWCNTs; for instance, concerning the presence of the density of state close to the Fermi level, the metallic tubes have a higher reactivity than the semiconducting ones,41 and simultaneously, the σ-π mixing caused by high curvature obviously affects the band structures of SWCNTs in this band region.42,43 Figure 5 shows the cycloadditions of CH2, NH, and SiH2 on the sidewall of the (6,2) and (6,3) tubes; a similar rule found in the cycloaddition of O is also observed. The boundary between the formations of adduct types I and II is close to K ) 1.5 nm-1. Here, the reason only the (6,2) and (6,3) tubes are considered is that, according to the results from the cycloaddition of the O atom, the boundary for the formations of two types of C-C bonds is expected to be between the (6,2) and (6,3) tubes. For the addition of the O atom, there are two type I and one type II structures for the (6,2) tube, whereas there are one type I and two type II structures for the (6,3) tube. It must be pointed out that when SiH2 is added to the sidewall of the (6,2) tube, only one C-C bond is broken, and the corresponding Eb is specially (41) Strano, M. S.; Dyke, C. A.; Usrey, M. L.; Barone, P. W.; Allen, M. J.; Shan, H.; Kittrell, C.; Hauge, R. H.; Tour, J. M.; Smalley, R. E. Science, 2003, 301, 1519-1522. (42) Gu¨lseren, O.; Yildirim, T.; Ciraci, S. Phys. ReV. B 2002, 65, 153405/ 1-4. (43) Zo´lyomi, V.; Ku¨rti, J. Phys. ReV. B 2004, 70, 085403/1-8.
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Figure 6. Variation in the difference in binding energy (∆Eb) between the formations of two types of adducts as a function of bond curvature for the cycloaddition of the O atom on some zigzag SWCNTs with large radii.
small (see the Supporting Information, Table S3). This may be related to the singlet ground state (The other X species except SiH2 and CCl2 are in the triplet ground states). 3.2. X/SWCNT System with Large or Small R. From Tables 1 and 2, it can be seen that, for the cycloadditions on the SWCNTs with moderate R values, the C-C bonds with θ ) 30° in the armchair tube or θ ) 0° in the zigzag tube are not destroyed, whereas those bonds with θ ) 90° in the armchair tube or θ ) 60° in the zigzag tube are broken. This conclusion is in agreement with Chen’s calculation results.22 However, for the SWCNTs with R < 0.2 nm and R > 0.7 nm, some attention must be paid to the cycloadditions on the C-C bonds with θ ) 30° in the armchair tubes and θ ) 60° in the zigzag tubes. Our results (see the Supporting Information, Tables S1 and S2) show that, for the C-C bonds of the armchair tubes with R < 0.2 nm, such as the O/(2,2) and O/(3,3) systems, it is easy to form the open structures even for the C-C bond with θ ) 30°; for the zigzag tubes with R > 0.7 nm, such as the O/(20,0) and O/(22,0) systems, it is also possible to form the 3MR structures for the C-C bond with θ ) 60°. It must be pointed out that, for the cycloaddition on SWCNTs with larger or smaller R values, the boundary of K should be properly extended. For instance, the competition between the structures of the type I and II adducts formed on the (3,3) armchair tube or the zigzag tubes from (10,0) to (22,0) may occur for 0.9 nm-1 < K < 1.9 nm-1 (see the Supporting Information, Table S1 or S2). The structural ambiguity in this K region may be related to the distortion of the SWCNTs with these sizes. Figure 6 displays the variation in the difference of binding energies (∆Eb) between the formations of two types of structures from (10,0) to (22,0) SWCNTs, and the positive value of ∆Eb means the formation of the type I adduct is favorable. The middle point of the K interval is close to 1.5 nm-1, and a similar result is also obtained for the periodic zigzag tube (see the Supporting Information, Figure S2). From the calculated results for the CCl2/SWCNT system (see the Supporting Information, Table S4), it also can be seen that the experimental observation26 that the local structure of CCl2/SWCNT is 3MR occurs in any bond of the zigzag SWCNTs (n,0) with smaller K (n g 20) and
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maybe in that in the middle of the competition region K. Because of the especially large K of the C-C bonds participating in the reaction, the theoretical prediction derived from the calculations for the CCl2 cycloadditions on the (9,0) and (10,0) tubes led to open structures.34 According to the K effect, for the armchair or chiral SWCNT, if only the 3MR configuration is obtained for the cycloaddition of CCl2, it requires a tube with a very large diameter, which is extremely unstable.
linearly with K. Several criteria, such as C-C bond strength (or length), tube diameter, and π-orbital misalignment angle, are not appropriate for determining the structure of the adduct because of various deficiencies. Our results indicate that only for the zigzag tube with smaller K will the cycloaddition of CCl2 on any C-C bond lead to the adducts with 3MR structure, which clarifies the contradiction between the experimental phenomenon26 and previous theoretical predictions.24, 34
Conclusions
Acknowledgment. We gratefully acknowledge financial support from the Natural Science Foundation of China (20273013, 20303002) and the Key Project of the Fujian Province (2005HZ01-2-6).
Bond curvature K proposed in this work can well-describe the local structures of the [2+1] cycloaddition for various C-C bonds of both the same and different SWCNTs, including the chiral, armchair, and zigzag SWNTs. The boundary value of K for the formation of the opened or 3MR adduct is approximately equal to 1.5 nm-1 for the tube with moderate R, and the values of Eb(I) and ∆RC-C(II) increase
Supporting Information Available: Derivation of the bondcurvature K and some detail results about the [2+1] cycloaddition. This material is available on the Web at http://www.pubs.acs.org. CM060563V
